Page 223 - A Course in Linear Algebra with Applications

7.1: Scalar Products in Euclidean Space 207

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It remains to define orthogonality in C . Two vectors X
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and Y in C are said to be orthogonal if
X*Y = 0.

We now make the blanket assertion that all the results estab-
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lished for scalar products in R carry over to complex scalar
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products in C . In particular the Cauchy-Schwartz and Tri-
angle Inequalities are valid.

Exercises 7.1

2
(~ \ ( x
1. Find the angle between the vectors I 4 I and I —2
V 3 / V 3.
2. Find the two unit vectors which are orthogonal to both of

the vectors I 3 J and I 1
1
\ - i / V .
3. Compute the vector and scalar projections of
on
"!) (i.

4. Show that the planes x — 3y + 4z = 12 and 2x — 6y + 8z = 6
are parallel and then find the shortest distance between them.
2
( \ f°\

5. If X = I —1 J and Y = 4 , find the vector product
V 3/ W

X x Y. Hence compute the area of the parallelogram whose
vertices have the following coordinates: (1, 1, 1), (3, 0, 4),
(1, 5, 3), (3, 4, 6).
6. Establish the following properties of the vector product:
(a) X x X = 0; (b) X x (Y + Z) = X x Y + X x Z;
(c)XxY = -YxX; (d) Xx(cY) = c{XxY) = (cX)xY.